526 vinculum

.

Then:

Using the double-angle formula for cosine, this can be

rewritten:

still with nradical terms. Put . Since

and , this gives:

and since, according to the

SQUEEZE RULE

,

as x→0, Viète’s formula follows.

See also

PI

, W

ALLIS

’

S PRODUCT

.

vinculum In the 15th century it was popular to place

a horizontal line above, or sometimes below, a group of

terms in an expression to indicate that those terms were

to be treated as a unit in the evaluation of that expres-

sion. The horizontal line was called a vinculum. Today

we use parentheses to indicate the order of operations.

For example,

x–——

y+ z= x– (y+ z)

Use of the vinculum first appeared in the 1484

manuscript Le triparty en la science des nombres (Tri-

party in the science of numbers) written by French

physician Nicola Chuquet (1445–88). As printing

presses were developed, parentheses were adopted so as

to ease typesetting.

Today a vinculum is primarily used to indicate

repeating decimals (for example, = with

the quantity 142857 regarded as a repeating unit) and to

denote the complex conjugate of a

COMPLEX NUMBER

.

The horizontal bar used in writing fractions is also

sometimes called a vinculum. It too can be thought of

as a device for indicating which terms are to be treated

as a unit. For example, in the expression the

entire quantity “x+ y” is to be divided by three, not x

alone nor yalone. Thus = + .

volume Loosely speaking, the volume of a three-

dimensional object or region is “the amount of space it

occupies.” Such a definition appeals to intuitive under-

standing. In general, however, it is very difficult to

explain precisely just what it is we mean by “space”

and the “amount” of it occupied. As with the notion of

AREA

, this is a serious issue, as demonstrated by the

Banach-Tarski paradox that arises in the careful study

of that topic.

In any case, given some kind of understanding of

what we mean by “area,” it is natural to then define the

“volume” of a simple object with vertical walls of height

hand a base of area Ato simply be the product A×h:

volume = area of base ×height

For example, a rectangular box aunits wide, bunits

deep, and cunits high has a base of area a×band thus

a volume given by a×b×c. A cylinder of height hwith

y

–

3

x

–

3

x+ y

–––

3

x+ y

–––

3

0.142857

1

–

7